Multipole Expansion Coefficients Research Articles

Recursive computation of the multipole expansions of layer potential integrals over simplices for efficient fast multipole accelerated boundary elements

In boundary element methods (BEM) in R3, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov subspace methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - they are only needed in terms of their contribution to the multipole expansion coefficients. We propose a new fast method - Quadrature to Expansion (Q2X) - for the analytical generation of the multipole expansion coefficients produced by the integral expressions for single and double layers on surface triangles; charge distributions over line segments and over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on the O(1) per moment cost recursive computation of the moments. The method is developed for boundary element methods involving the Laplace Green's function in R3. The derived recursions are first compared against classical quadrature algorithms, and then integrated into FMM accelerated boundary element and vortex element methods. Numerical tests are presented and discussed.

Rotational Excitation of NCCN by p-H2(jc = 0) at Low Temperatures

NCCN is a potential candidate for detection in interstellar space, but to model its emission spectra, accurate collisional rates are required. Here, we present a first truncated ab initio four-dimensional potential energy surface of NCCN–H2, generated using explicitly correlated coupled-cluster singles and doubles with the perturbative triple excitations [CCSD(T)-F12b] method and aug-cc-pVTZ basis set. The surface allows NCCN to rotate in the x–z plane, and H2 is restricted to three orientations [(a) parallel to the z axis, (b) perpendicular to the z axis but still in the x–z plane, and (c) perpendicular to the z axis and also perpendicular to the x–z plane] and averaged to obtain a reduced two-dimensional (2D) surface. To augment data points, the surface is fitted into a neural networks model. Multipole expansion coefficients for this reduced 2D surface of NCCN–H2 have been calculated and utilized in determining collisional cross sections using the close-coupling method for collision energies until 1500 cm–1. Finally, the rate coefficients for reduced NCCN–H2 have been computed. These rate coefficients closely approximate the collision of NCCN with p-H2(jc = 0), and they are greater by 1.8–2.8 times than the rates of NCCN–He previously investigated.

Quasinormal modes of spheroidal resonators in the null-field framework and application to hybrid anapoles

A model for calculating the quasinormal modes (QNMs) of homogeneous and core-shell spheroidal resonators is formulated in the null-field framework. Relying on a multipole expansion of the electromagnetic fields, the Mittag-Leffler theorem is employed to extract the QNMs from the poles of a response matrix. Physical quantities such as the extinction and scattering efficiency and the total internal energy of a mode are expressed directly in terms of the multipole expansion coefficients. We apply the model to study hybrid anapoles, which are radiationless states characterized by an enhancement of the total internal energy. Interference between excited QNMs and the background comprising of all other QNMs leads to a suppression of the extinction efficiency. Further, we show this suppression is due to Fano-like interference between resonant and background multipoles.

Quantitative assessment of finite-element models for magnetostatic field calculations

We present quantitative means for assessing the numerical accuracy of static magnetic field calculations in finite-element models. Our calculations use the three-dimensional Opera simulation software suite of Dassault Systèmes. Our need to assess the effects of fringe fields requires such a 3D algorithm. While we do discuss and compare our approach to a method of accuracy estimation used in the Opera post-processor, our methods are generally applicable to any model using relaxation techniques in finite-element systems. For purposes of illustration, we present modeling and analysis of two types of quadrupole electromagnets presently in operation in the south arc of the Cornell Electron Storage Ring (CESR). Calculations of field multipole expansion coefficients and numerical deviations from Maxwell’s equations in source-free regions are discussed, with emphasis on the dependence of their accuracy on changes to the finite-element model. Successive refinement steps in the finite-element model for the non-extraction type of CESR south arc quadrupole achieve a reduction in the RMS value of the longitudinal component of the curl vector on the magnet axis by a factor of nearly 70 from 4.38×10−2 T/m to 6.36×10−4 T/m, which is 0.0021% of the field gradient. An accuracy of 2.9% is achieved for a dodecapole coefficient of 2.4×10−4 of that of the quadrupole in the longitudinal integral of the vertical field gradient at a radius of 1 cm.

Acceleration of unsteady vortex lattice method via dipole panel fast multipole method

The Unsteady Vortex Lattice Method (UVLM) is a medium-fidelity aerodynamic tool that has been widely used in aeroelasticity and flight dynamics simulations. The most time-consuming step is the evaluation of the induced velocity. Supposing that the number of bound and wake lattices is N and the computational cost is ON2, we present an ON Dipole Panel Fast Multipole Method (DPFMM) for the rapid evaluation of the induced velocity in UVLM. The multipole expansion coefficients of a quadrilateral dipole panel have been derived in spherical coordinates, whose accuracy is the same as that of the Biot-Savart kernel at the same truncation degree P. Two methods (the loosening method and the shrinking method) are proposed and tested for space partitioning volumetric panels. Compared with FMM for vortex filaments (with three harmonics), DPFMM is approximately two times faster for N∈[103,106]. The simulation time of a multirotor (N∼104) is reduced from 100 min (with unaccelerated direct solver) to 2 min (with DPFMM).

Simulation of Sound Propagation Over an Infinite Impedance Plane by Using a Fast Multipole BEM

In this paper, a half-space fast multipole BEM is developed for the simulation of three-dimensional acoustic problems above an infinite impedance plane. The half-space impedance Green’s function involving a complex line source is used, so that both mass-like and spring-like impedance boundary conditions on the infinite plane can be explicitly satisfied and the infinite plane is not required to be discretized. The Burton–Miller method is employed to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Image relations of the multipole expansion coefficients are used and the half-space impedance Green’s function is modified to apply such relations to avoid calculating, translating and saving the multipole/local expansion coefficients in the image domain. An automatic integrator with adaptive interval subdivision is further adopted to calculate the line integral contained in the M2L translation formula accurately and efficiently. Numerical examples are given to show the validity and potential of the method.

Acoustic scattering and forces on an arbitrarily sized fluid sphere by a general acoustic field

In this work, we present a solution to the acoustic scattering problem by a fluidic sphere. The method is based on multipole expansion coefficients for a general acoustic field, which is applied to two common acoustic fields: stationary waves and arbitrarily located Bessel beams of orders m = 0 and m = 1. We present a new solution for the acoustic radiation force in terms of pressure and velocity instead of velocity potentials that are in agreement with these formulations. The behavior of the acoustic force and scattering is in agreement to the Rayleigh, Mie–like and Geometric scattering regimes. Results for force in the stationary pressure waves, typical for acoustic levitation trap, show nodes where small particles can be trapped by a restoring acoustic force. For the acoustic Bessel beam considered, the resulting force vectors point towards the corresponding intensity rings.

Resonance Scattering of an Arbitrary Bessel Beam by a Spherical Object.

In our prior work, multipole expansion coefficients of a Bessel beam of arbitrary order with respect to an object of arbitrary location was derived analytically and combined with the T-matrix method to numerically compute the scattering. The present work is extended to directly use the multipole expansion to calculate the scattering from an elastic spherical target. The incident Bessel beam is located at an arbitrary location with respect to the center of the sphere. The far-field resonance scattering in both on- and off-axis incidences is calculated by subtracting an appropriate background from the total scattering based on the resonance scattering theory. The results reveal that the ordinary or helicoidal Bessel beams with selected half-cone angle could lead to resonance suppression at some resonance frequencies for on-axis incidence, however, not for off-axis incidence. The intrinsic property of the elastic scattering contribution from a sphere is shown by the three-dimensional patterns of resonance scattering. The resonance scattering and suppression from thin and thick shells are also noted.

Multipole accelerated magnetic field calculations for superconducting circuits

This paper presents an adaptive fast multipole algorithm to accelerate the computation of magnetic fields surrounding current-carrying superconducting volumes. The algorithm is based on a hierarchical tree of cubic cells and vector spherical harmonics are used to approximate the gradient of Green’s function. The derivation of the multipole and local expansion coefficients are presented, including the implementation of these coefficients within FastCap’s multipole algorithm. We demonstrate how the proposed algorithm can be used to calculate the magnetic fields of trapped flux and the magnetic fields surrounding type-II superconducting microstrips and compare the results to analytical solutions. The overall complexity of our multipole algorithm is found empirically to rise linearly, , with N evaluation points.

The fast multi-pole indirect BEM for solving high-frequency seismic wave scattering by three-dimensional superficial irregularities

Taking full advantage of the indirect boundary element method (IBEM) and fast multi-pole expansion algorithm, this paper proposes a fast multi-pole indirect boundary element method (FM-IBEM) to solve the scattering of high-frequency seismic waves by three-dimensional (3-D) superficial irregularities or heterogeneity in a solid half-space. First, IBEM utilizes an exact dynamic Green's function for a full-space to construct the scattered wave field. Subsequently, by employing plane waves expansion of 3-D potential functions of compressional and shear waves, the multi-pole expansion and local expansion coefficients were derived. Implementation of FM-IBEM is presented in detail for wave-scattering problems. Numerical examples illustrate that the proposed FM-IBEM can reduce the memory required by more than an order of magnitude and also greatly improve the computing efficiency, retaining excellent accuracy as well. Ultimately, several high-frequency plane wave scattering problems of 3-D superficial irregularities in a solid half-space are illustrated, and several important scattering characteristics are described based on the high-precision numerical results.

A Fast and Accurate Computational Method for Multipole Expansion Coefficients in the Integral Equation Method

A Fast and Accurate Computational Method for Multipole Expansion Coefficients in the Integral Equation Method

Optimization of RF multipole ion trap geometries

Radio-frequency (rf) traps are ideal places to store cold ions for spectroscopic experiments. Specific multipole configurations are suited best for different applications but have to be modified to allow e.g. for a proper overlap of a laser beam waist with the ion cloud. Therefore the corresponding trapping fields should be shaped accordingly. To achieve this goal highly accurate electrical potentials of rf multipole traps and the resulting effective trapping potentials are calculated using the boundary element method (BEM). These calculations are used to evaluate imperfections and to optimize the field geometry. For that purpose the complex fields are reduced to a small set of multipole expansion coefficients. Desirable values for these coefficients are met by systematic changes of real trap dimensions from CAD designs. The effect of misalignment of a linear quadrupole, the optimization of an optically open Paul trap, the influence of steering electrodes (end electrode and ring electrode) on a 22-pole ion trap and the effect of the micro motion on the lowest reachable temperatures in such a trap are discussed.

Optical Fano resonances in a nonconcentric nanoshell.

The interaction of light with a metal nanoshell with an off-center core generates multipoles of all orders. We show here that the matrix elements used to compute the multipole expansion coefficients can be derived analytically and, with this result, we can show explicitly how the dipole and quadrupole terms in the expansion are coupled and give rise to a Fano resonance. We also show that the off-center core significantly increases the electric field enhancement at the shell surface compared to the concentric case, which can be exploited for surface-enhanced sensing. The multipole solutions are confirmed with finite-element calculations.

Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam

In this paper, we derive the analytical expression for the multipole expansion coefficients of scattering and interior fields of a graphene-coated dielectric particle under the illumination of an arbitrary optical beam. By using this arbitrary beam theory, we systematically investigate the optical forces exerted on the graphene-coated particle by a focused Gaussian beam. Via tuning the chemical potential of the graphene, the optical force spectra could be modulated accordingly at resonant excitation. The hybridized whispering gallery mode of the electromagnetic field inside the graphene-coated polystyrene particle is more intensively localized than the pure polystyrene particle, which leads to a weakened morphology-dependent resonance in the optical forces. These investigations could open new perspectives for dynamic engineering of optical manipulations in optical tweezers applications.

The multi-domain FMM-IBEM to model elastic wave scattering by three-dimensional inclusions in infinite domain

The scattering of elastic wave by three-dimensional (3-D) inclusions embedded in solid infinite domain is solved by a Fast Multipole accelerated multi-domain Indirect Boundary Element Method (FMM-IBEM). Based on the single-layer potential theory, the scattered field is constructed by virtual uniform loads acted on circular elements covering the boundary surface, and then the analytic integration of the exact Green’s function for infinite space can be applied. By using the Taylor series expansion of compressional and shear waves potential function, the multipole expansion and local expansion coefficients of Green’s function are deduced. The detailed procedure of multi-domain FMM-IBEM for such problem is presented. It is verified that this method substantially improves the computing efficiency and reduces the memory storage, then the elastic wave scattering problems involving millions of degrees of freedom (DOFs) can be solved accurately and efficiently on an ordinary workstations. Finally, the scattering problems of P, SV waves by spherical inclusions group and 3-D random inclusions in elastic infinite domain are solved, and some important characteristics of plane P and SV waves scattering by inclusions group are discussed through typical numerical results.

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/plane-symmetric acoustic wave problems. The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only. Moreover, a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived, and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating, translating and saving the multipole/local expansion coefficients of the image domain. The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems. As for exterior acoustic problems, the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Details on the implementation of the present method are described, and numerical examples are given to demonstrate its accuracy and efficiency.

Contribution from cosmological scalar perturbations to the angular velocity spectrum of extragalactic sources

The possibility of the influence of adiabatic scalar perturbations on the angular velocity spectrum of extragalactic sources is considered. The multipole expansion coefficients of the angular velocity field in terms of vector spherical harmonics are calculated. We show that there is no contribution from adiabatic perturbations to the angular spectrum for a spatially flat Universe at the dusty stage, while there is a contribution only to the electric multiple coefficients at the stage of Λ-term domination. The cases of long-wavelength and short-wavelength perturbations are considered separately. The relationship between the multipole angular velocity spectrum and the primordial scalar perturbation spectrum is discussed.

Theory of magneto-optical effects in nanostructured ferromagnetics

We present a theory of light interaction with a magnetic nanostructured material consisting of a periodic array of ferromagnetic nanocylinders magnetized along the symmetry axis. Using a multiple scattering approach, we calculated the amplitudes of diffracted waves in terms of multipole expansion coefficients and used the developed formalism to describe the experimentally observed enhancement of the magneto-optical effects in magnetic nanostructures. The analysis provides a physical interpretation of the enhancement in terms of interference between the resonant and nonresonant scattering processes. The resonant scattering is caused by excitation of dipole surface-plasmon eigenmodes; the nonresonant one is associated with the monopole modes.

Efficient technique in low-frequency fast multipole boundary element method for plane-symmetric acoustic problems

The fast multipole boundary element method (FMBEM) has been well known as a highly efficient BEM with the use of the fast multipole method (FMM). In the present paper, an efficient technique for plane-symmetric acoustic problems is proposed in the framework of an FMBEM based on the original multipole expansion theory (FMBEM for low-frequency problems: LF-FMBEM). Presented here are concrete computational procedures, which are based on the symmetries among multipole expansion coefficients for a plane-symmetric sound field produced by monopole or dipole sources. The proposed technique is straightforwardly applicable to a variety of formulations for the BEM, such as hypersingular, Burton–Miller, and indirect formulations. Numerical results show an ideal improvement of computational efficiency, with the proposed technique reducing both the computation time and required memory to about 1/2nsym of those using the standard LF-FMBEM, where nsym is the number of planes of symmetry.

Revision of FMM–Yukawa: An adaptive fast multipole method for screened Coulomb interactions

FMM–Yukawa is a mathematical software package primarily for rapid evaluation of the screened Coulomb interactions of N particles in three dimensional space. Since its release, we have revised and re-organized the data structure, software architecture, and user interface, for the purpose of enabling more flexible, broader and easier use of the package. The package and its documentation are available at http://www.fastmultipole.org/, along with a few other closely related mathematical software packages.